Multiplications of maximal rank in the cohomology of $${\mathbb{P}^1\times \mathbb{P}^1}$$

Abstract

Let Q = P 1 × P 1 and let C ⊆ Q be a curve of type (a, b) having equation F = 0. The main purpose of this paper is to analize the multiplicative structure of the bi-graded module H 1 * O Q , in particular to prove that for any r, s ≥ 0 the multiplication mapinduced by F has maximal rank for the general C of type (a, b). Interpretations of this problem in the contexts of multilinear algebra and differential algebra are emphasized.